Verify that the standard error falls as the square root of n
Three steps, the way the exam actually works: work through the lab, write down your own measurements, then answer a 9-point free response. What you recorded goes to the grader with your writing, so a conclusion that does not follow from your own numbers will cost you the point — exactly as it would with a real reader.
Predict before you look
- The standard deviation of the sampling distribution of x̄ is σ/√n. From that formula alone, predict what happens to it when n is multiplied by four.
- The lab reports a "predicted SE" alongside the observed spread of the sample means. One of those two numbers is computed from a formula and one from random draws — say which before you begin.
Nothing to submit here — these are to think through, so the prediction below is an informed one rather than a guess.
Commit to an answer now. It is not graded and being wrong costs nothing — the point is to have something specific to reconcile against once you have the data.
Answer every prediction to unlock the lab. A sentence is enough.
Run the investigation
Predictions first
The procedure and the simulation unlock once you have committed above. Observing before predicting is how a wrong intuition survives a lab intact.
Record what you measured
These are your numbers, not ours. The grader sees them, so your conclusions have to follow from what you actually recorded.
| σ reported for the Uniform population | |
|---|---|
| μ reported for the Uniform population | |
| Predicted SE, Uniform, n = 5 | |
| Predicted SE, Uniform, n = 10 | |
| Predicted SE, Uniform, n = 20 | |
| Predicted SE, Uniform, n = 50 | |
| Ratio of the n = 5 reading to the n = 20 reading(Divide your two recorded values.) | |
| σ reported for the Normal population | |
| Predicted SE, Normal, n = 25 |
Answer the free response
Using your recorded values: (a) Show by direct calculation that your n = 5 reading equals σ/√n for the Uniform population. (b) State the ratio you computed between the n = 5 and n = 20 readings and explain why it is exactly 2 rather than 4. (c) Determine how large n would have to be to bring the Uniform population's standard error below 0.20, and show your work. (d) Explain why your two Normal-population readings are smaller than the corresponding Uniform ones, and state what that implies about how much data a study needs.
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